I mean this in the broadest of ways...

Some musts for me...

- bring an actual digital camera (not phone/tablet) since it gives me more flexibility flexibility in capturing images

- plastic wrap around bike box or case to hold everything together

- a roller gizmo to better manoeuvre a bike box.

Mine is "bring the bead jack to prevent flats, especially in the rain." 

Most of the tires I've used for touring are pretty hard to get on, so years ago I bought a Kool Stop bead jack tool for use at home and on the road. The one time I forgot to carry it on a tour, I got my only flat of the year - just as it started to rain.  Wasn't too bad - I was near one of those country school bus shelter/sheds and stayed dry as I grunted and groaned and finally got the tire back on.

I hate to tempt the flat gods, but I have never had a flat while carrying that plastic bead jack.

I had one of those plastic tire jacks (I forget what brand) and managed to break it putting on a particularly stubborn Schwalbe Marathon tire. So my latest purchase is a set of steel core tire levers from Park Tool. Haven’t had to use them yet but hopefully they’ll do the trick when the time inevitably comes.

For our forthcoming Australia tour ..... camping chairs (at last).

This is a small item, but we used it nearly every day. A Sawyer Mini water filter to fill our water bottles and keep us from buying water and thus discarding yet another piece of plastic that may or may not get recycled. Of course if you are in an area where tap water is safe to drink this isn't needed. 

https://www.amazon.com/Sawyer-Products-Water-Filtration-System/dp/B00TOX6UM6?th=1

This is an intriguing question. Do I do anything clever? Hmmm. 

Something that has worked for me on numerous occasions is to call our lodging the night before, and in my best 'could you please help me' tone of voice, explain that we are touring cyclists, who ride early in the day to beat the heat (or rain, or traffic - whatever sounds most convincing), and if at all possible, could we check into our room early. I quickly add in if it seems appropriate, that I realize it's the summer season and they are probably short staffed, but if there was at all a chance . . . . with my voice trailing off into a hopeful silence. 

My daughter listened to my spiel once and was horrified at what she considered to be groveling.  I think it is human nature to want to help, and if you ask nicely, people usually do help.

Something we've carried the last few years that has been helpful, is a small pair of scissors. This past summer Jacinto used them to cut the fingers off of a pair of gardening gloves when he couldn't find a replacement pair of short fingered cycling gloves. 

Wwe absolutely bring our metal tire levers. The plastic ones are useless.

Same for us so now we  ALWAYS carry metal tire levers.

cannot remember now if it was you who recommended the Sawyer mini filter, but we bring one  on every trip to Mexico and it has both saved us money and saved the planet from some unessessary plastic waste.

I have to admit that I tend to use the "I know it's early and before check in time, but..." and use that same trailing dot-dot-dot that you described. If the room is ready, it seems to work. So nice to be able to drop the bike and gear to head out into town to explore on foot!